On entire functions restricted to intervals, partition of unities, and dual Gabor frames

TL;DR

This paper characterizes entire functions that generate partitions of unity when restricted to intervals and explores their application in constructing dual Gabor frames with low redundancy and specific regularity.

Contribution

It provides a characterization of entire functions forming partitions of unity via integer translates and applies this to construct dual Gabor frames with small support and desired regularity.

Findings

Characterization of entire functions leading to partitions of unity.

Construction of dual Gabor frames with low redundancy.

Generation of partitions of unity with functions of small support.

Abstract

Partition of unities appear in many places in analysis. Typically they are generated by compactly supported functions with a certain regularity. In this paper we consider partition of unities obtained as integer-translates of entire functions restricted to finite intervals. We characterize the entire functions that lead to a partition of unity in this way, and we provide characterizations of the "cut-off" entire functions, considered as functions of a real variable, to have desired regularity. In particular we obtain partition of unities generated by functions with small support and desired regularity. Applied to Gabor analysis this leads to constructions of dual pairs of Gabor frames with low redundancy, generated by trigonometric polynomials with small support and desired regularity.